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Monte Carlo Methods and the Genetic Algorithm Definitions and Considerations

Monte Carlo Methods and the Genetic Algorithm Definitions and Considerations. John E. Nawn MAT 5900 March 17 th , 2011. What is the Genetic Algorithm?. Heuristic search method employing randomness in order to determine the optimal solution to a wide range of problems Applications include:

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Monte Carlo Methods and the Genetic Algorithm Definitions and Considerations

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  1. Monte Carlo Methods and the Genetic AlgorithmDefinitions and Considerations John E. Nawn MAT 5900 March 17th, 2011

  2. What is the Genetic Algorithm? • Heuristic search method employing randomness in order to determine the optimal solution to a wide range of problems • Applications include: • Economics • Number Theory • Rankings • Path Length Determination (TSP, etc.) • Based in Neo-Darwinian theory

  3. History of Genetic Algorithms • Operational Research (1940s and 1950s) – birth of heuristics • Evolutionsstrategie – Rechenberg and Schwefel (1960s) • Adaptation in Natural and Artificial Systems – John Holland (1975) • Increased computational complexity (1990s – 2000s)

  4. Evolution: A Survey • On the Origin of Species – Charles Darwin (1859) • Proposed natural selection – environment creates selection pressure for individuals in a species • Selected advantages may be heritable: provides method for determining fitness of offspring • What Darwin (and biologists) didn’t know…

  5. Genetics: A Survey • Gregor Mendel (1863) • Individuals within a species carry directions for their promulgation • Segregation (First Law) • Independent Assortment (Second Law) • Increasing technology and the discovery of mutations and crossovers • Genotype and phenotype

  6. Terminology • Population • Set of possible solutions in any given generation • Chromosomes • Basic units that undergo reproduction in the algorithm • Two types: binary and non-binary • Minimum size requirements • Genes and alleles • Reproduction

  7. Terminology • Mutation • Process of changing allele values in a chromosome • Inversions • How often? • What type? • Crossover • Process of combining parental chromosomes to yield new chromosomes • What type?

  8. Terminology • Selection • Criterion • Fitness functions • Reeves and Rowe: • Tournament selection • Ranking • Termination • Diversity thresholds • Generation limits • Computational limits

  9. Minimum String Length Requirements Reeves, Colin R.; p. 28

  10. Mutations • Simplicity of method • Binary • Reversal of alleles • Non-binary • Stochastic selection of new alleles • Differing mutation rates • Selecting complete mutations and error repair

  11. Crossovers (X) • Binary • NX – N-point crossovers • UX – Uniform crossover, or linear operator “masks” • Non-Binary • Difficulty in applying n-point crossovers • PMX – Partially matched crossover • UX – “in/out” order crossovers • Further possibilities – Fox/ McMahon and Poon/ Carter

  12. Fitness Functions • Method comparing gene success • Roulette wheel model of selection • Selection pressure = individual fitness/ total fitness • Benefit of larger selection pressure • Niches

  13. Critiques of the Genetic Algorithm:Biological and Philosophical Arguments • What is natural selection selecting for? • Evolution as a theory or fact: Lisa Gatlin • Individual genes and group interactions • Lamarckian or Darwinian evolution?

  14. Critiques of the Genetic Algorithm:Mathematical Arguments • Lack of theory in heuristic applications • Newton’s Method problem • Best possible solution or best solution? • Pseudo-randomness • Similarities to Markov chains and processes (a.k.a. t – 1 dependency)

  15. What to Expect Next • Crossover possibilities • Holland’s method - schemata approaches • Three applications: • General Path Problems or the Traveling Salesman Problem (TSP) • Ranking Styles • Stock Selection

  16. Selected Bibliography • Craig, Nancy L. et. al. Molecular Biology: Principles of Genome Function. New York: Oxford University Press, 2010. Print. • Krzanowski, Roman and Jonathan Raper. Spatial Evolutionary Modeling. New York: Oxford University, Inc., 2001. Print. • Reeves, Colin R. and Johathan E. Rowe. Genetic Algorithms: Principles and Perspectives: A Guide to GA Theory. Boston: Kluwer Academic Publishers, 2003. Print. • Russell, Peter J. iGenetics: A Mendelian Approach. San Francisco: Pearson Education, Inc., 2005. Print

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